Given: tan θ + cot θ = 5. We need to find tan²θ + cot²θ.

Recall that cot θ = 1/tan θ. Let t = tan θ for simplicity. Then the given equation becomes t + 1/t = 5.

Square both sides: (t + 1/t)² = 5². This expands to t² + 2(t)(1/t) + (1/t)² = 25, which simplifies to t² + 2 + 1/t² = 25.

Therefore, t² + 1/t² = 25 - 2 = 23. Since t² = tan²θ and 1/t² = cot²θ, the answer is 23.

Final Answer: 23